920
The smallest squares containing k 920's :
39204 = 1982,
9792092025 = 989552,
592060920619204 = 243323022.
9202± 3 are primes.
9202 = 846400 appears in the decimal expression of π:
π = 3.14159•••846400••• (from the 79090th digit).
by Yoshio Mimura, Kobe, Japan
921
The smallest squares containing k 921's :
7921 = 892,
1921857921 = 438392,
64921759219216 = 80574042.
The squares which begin with 921 and end in 921 are
92166280921 = 3035892, 921290905921 = 9598392, 921429127921 = 9599112,
921770887921 = 9600892, 921909145921 = 9601612,...
9212 = 848241, a zigzag square.
(22 + 9)(42 + 9)(62 + 9)(72 + 9) = 9212 + 9.
18k + 282k + 921k + 1380k are squares for k = 1,2,3 (512, 16832, 585812).
6k + 408k + 690k + 921k are squares for k = 1,2,3 (452, 12212, 343172).
3-by-3 magic squares consisting of different squares with constant 9212:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(7, 176, 904, 496, 761, 152, 776, 488, 89), | (16, 268, 881, 463, 764, 224, 796, 439, 148), |
(16, 359, 848, 544, 688, 281, 743, 496, 224), | (20, 175, 904, 604, 680, 145, 695, 596, 100), |
(20, 271, 880, 329, 820, 260, 860, 320, 79), | (44, 128, 911, 244, 881, 112, 887, 236, 76), |
(44, 337, 856, 649, 596, 268, 652, 616, 209), | (49, 224, 892, 304, 847, 196, 868, 284, 119), |
(49, 356, 848, 512, 716, 271, 764, 457, 236), | (54, 387, 834, 477, 726, 306, 786, 414, 243), |
(55, 460, 796, 596, 625, 320, 700, 496, 335), | (56, 167, 904, 391, 824, 128, 832, 376, 121), |
(56, 431, 812, 623, 616, 284, 676, 532, 329), | (64, 356, 847, 572, 649, 316, 719, 548, 176), |
(76, 247, 884, 559, 716, 152, 728, 524, 209), | (76, 457, 796, 568, 604, 401, 721, 524, 232), |
(79, 496, 772, 548, 596, 439, 736, 497, 244), | (89, 328, 856, 632, 601, 296, 664, 616, 167), |
(92, 364, 841, 401, 776, 292, 824, 337, 236), | (99, 246, 882, 378, 819, 186, 834, 342, 189), |
(119, 524, 748, 596, 527, 464, 692, 544, 271), | (145, 296, 860, 440, 785, 196, 796, 380, 265), |
(167, 376, 824, 616, 596, 337, 664, 593, 236), | (188, 544, 719, 631, 604, 292, 644, 433, 496), |
(236, 593, 664, 628, 376, 559, 631, 596, 308) |
9212 = 848241, 8 + 48 + 24 + 1 = 92,
9212 = 848241, 848 + 241 = 332.
by Yoshio Mimura, Kobe, Japan
922
The smallest squares containing k 922's :
99225 = 3152,
19792269225 = 1406852,
595729229229225 = 244075652.
(473 / 922)2 = 0.263184579... (Komachic).
9222 = 850084, 82 + 52 + 02 + 02 + 82 + 42 = 132.
9222 = 850084, 8 + 5 + 0 + 0 + 8 + 4 = 52,
9222 = 850084, 82 + 52 + 02 + 02 + 82 + 42 = 132,
9222 = 850084, 85 + 0 + 0 + 84 = 132.
by Yoshio Mimura, Kobe, Japan
923
The smallest squares containing k 923's :
799236 = 8942,
5092392321 = 713612,
192392379231364 = 138705582.
923 is the second integer which is the sum of a square and a prime in 12 ways:
22 + 919, 42 + 907, 62 + 887, 82 + 859, 102 + 823, 142 + 727, 182 + 599, 202 + 523, 222 + 439, 242 + 347, 282 + 139, 302 + 23.
9232 = 1212 + 4022 + 8222 : 2282 + 2042 + 1212 = 3292.
(13 + 23 + ... + 333)(343 + 353 + ... + 7143)(7153 + 7163 + ... + 9233) = 489149663282282.
10k + 164k + 362k + 833k are squares for k = 1,2,3 (372, 9232, 250972).
3-by-3 magic squares consisting of different squares with constant 9232:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(14, 498, 777, 642, 553, 366, 663, 546, 338), | (39, 282, 878, 338, 822, 249, 858, 311, 138), |
(39, 338, 858, 418, 759, 318, 822, 402, 121), | (39, 338, 858, 598, 663, 234, 702, 546, 247), |
(39, 598, 702, 642, 522, 409, 662, 471, 438), | (42, 446, 807, 546, 663, 338, 743, 462, 294), |
(57, 266, 882, 546, 702, 247, 742, 537, 114), | (58, 294, 873, 567, 678, 266, 726, 553, 138), |
(90, 455, 798, 490, 702, 345, 777, 390, 310), | (94, 327, 858, 438, 774, 247, 807, 382, 234), |
(102, 423, 814, 473, 726, 318, 786, 382, 297), | (105, 498, 770, 590, 630, 327, 702, 455, 390), |
(114, 402, 823, 633, 634, 222, 662, 537, 354), | (130, 423, 810, 585, 590, 402, 702, 570, 185), |
(231, 498, 742, 558, 679, 282, 698, 378, 471), | (234, 558, 697, 598, 633, 306, 663, 374, 522) |
9232 = 851929, 82 + 52 + 12 + 92 + 22 + 92 = 162.
Page of Squares : First Upload January 30, 2006 ; Last Revised August 17, 2013by Yoshio Mimura, Kobe, Japan
924
The smallest squares containing k 924's :
13924 = 1182,
8069249241 = 898292,
392492402039241 = 198114212.
The squares which begin with 924 and end in 924 are
92487757924 = 3041182, 924255349924 = 9613822, 924709177924 = 9616182,
9240882573924 = 30398822, 9242317453924 = 30401182,...
9242 = 103 + 423 + 923.
9242 = 853776 , 853 + 77 - 6 = 924.
The integral triangle of sides 1617, 2425, 3944 (or 693, 11986, 12665) has square area 9242.
924k + 1078k + 1617k + 2310k are squares for k = 1,2,3 (772, 31572, 1363672).
(1 + 2)(3 + 4 + ... + 30)(31 + 32 + ... + 46) = 9242,
(1 + 2 + ... + 32)(33 + 34 + ... + 65) = 9242.
9242 = 853776, 8 + 5 + 3 + 7 + 7 + 6 = 62,
9242 = 853776, 8 + 53 + 7 + 7 + 6 = 92,
9242 = 853776, 8 + 53 + 7 + 76 = 122,
9242 = 853776, 8 + 53 + 77 + 6 = 122.
by Yoshio Mimura, Kobe, Japan
925
The smallest squares containing k 925's :
925444 = 9622,
9259250625 = 962252,
154925792592561 = 124469192.
9252 = (22 + 1)(62 + 1)(682 + 1).
(388 / 925)2 = 0.175946238... (Komachic).
13 - 23 + 33 - 43 + 53 - ... - 9243 + 9253 = 199092.
3012 + 3022 + 3032 + ... + 9252 = 159752,
2042 + 2052 + 2062 + ... + 9252 = 161692.
3-by-3 magic squares consisting of different squares with constant 9252:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 259, 888, 300, 840, 245, 875, 288, 84), | (0, 285, 880, 555, 704, 228, 740, 528, 171), |
(0, 300, 875, 555, 700, 240, 740, 525, 180), | (0, 520, 765, 555, 612, 416, 740, 459, 312), |
(3, 96, 920, 600, 700, 75, 704, 597, 60), | (11, 360, 852, 648, 605, 264, 660, 600, 245), |
(16, 213, 900, 387, 816, 200, 840, 380, 75), | (24, 93, 920, 380, 840, 75, 843, 376, 60), |
(24, 268, 885, 632, 651, 180, 675, 600, 200), | (24, 515, 768, 565, 600, 420, 732, 480, 299), |
(52, 264, 885, 480, 765, 200, 789, 448, 180), | (60, 205, 900, 612, 684, 115, 691, 588, 180), |
(72, 420, 821, 605, 600, 360, 696, 565, 228), | (75, 200, 900, 504, 765, 128, 772, 480, 171), |
(75, 380, 840, 600, 660, 245, 700, 525, 300), | (75, 444, 808, 600, 592, 381, 700, 555, 240), |
(92, 420, 819, 525, 700, 300, 756, 435, 308), | (124, 360, 843, 600, 675, 200, 693, 520, 324), |
(180, 525, 740, 556, 660, 333, 717, 380, 444), | (192, 515, 744, 619, 480, 492, 660, 600, 245) |
9252 = 855625, 8 + 5 + 5 + 6 + 25 = 72,
9252 = 855625, 85 + 5 + 6 + 25 = 112.
by Yoshio Mimura, Kobe, Japan
926
The smallest squares containing k 926's :
292681 = 5412,
9269260729 = 962772,
392692641792676 = 198164742.
9262 = 857476, a zigzag square.
(471 / 926)2 = 0.258713946... (Komachic).
9262 = 857476, 8 + 5 + 7 + 4 + 76 = 102,
9262 = 857476, 8 + 5 + 74 + 7 + 6 = 102.
by Yoshio Mimura, Kobe, Japan
927
The smallest squares containing k 927's :
192721 = 4392,
392792761 = 198192,
19279273927489 = 43908172.
9272 = 64 + 214 + 244 + 244.
9272 + 9282 + 9292 + ... + 51352 = 2118532.
3-by-3 magic squares consisting of different squares with constant 9272:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 218, 901, 341, 838, 202, 862, 331, 82), | (2, 485, 790, 590, 610, 373, 715, 502, 310), |
(5, 230, 898, 610, 677, 170, 698, 590, 155), | (12, 303, 876, 372, 804, 273, 849, 348, 132), |
(12, 456, 807, 519, 672, 372, 768, 447, 264), | (22, 134, 917, 526, 757, 98, 763, 518, 94), |
(22, 229, 898, 502, 758, 181, 779, 482, 142), | (22, 322, 869, 581, 682, 238, 722, 539, 218), |
(26, 362, 853, 418, 757, 334, 827, 394, 142), | (33, 264, 888, 552, 708, 231, 744, 537, 132), |
(37, 386, 842, 538, 677, 334, 754, 502, 197), | (43, 226, 898, 394, 818, 187, 838, 373, 134), |
(43, 358, 854, 434, 763, 298, 818, 386, 203), | (43, 554, 742, 622, 533, 434, 686, 518, 347), |
(48, 420, 825, 600, 615, 348, 705, 552, 240), | (57, 516, 768, 636, 537, 408, 672, 552, 321), |
(82, 379, 842, 491, 698, 362, 782, 478, 139), | (98, 290, 875, 475, 770, 202, 790, 427, 230), |
(98, 293, 874, 347, 826, 238, 854, 302, 197), | (106, 373, 842, 523, 722, 254, 758, 446, 293), |
(111, 348, 852, 588, 687, 204, 708, 516, 303), | (133, 238, 886, 314, 853, 182, 862, 274, 203), |
(139, 538, 742, 638, 482, 469, 658, 581, 298), | (149, 358, 842, 562, 709, 202, 722, 478, 331), |
(166, 587, 698, 622, 446, 523, 667, 562, 314), | (182, 494, 763, 587, 658, 286, 694, 427, 442), |
(230, 523, 730, 565, 670, 302, 698, 370, 485) |
9272 = 859329, 8 + 5 + 9 + 3 + 2 + 9 = 62,
9272 = 859329, 8 + 59 + 3 + 2 + 9 = 92,
9272 = 859329, 859 + 32 + 9 = 302.
by Yoshio Mimura, Kobe, Japan
928
The smallest squares containing k 928's :
49284 = 2222,
2319289281 = 481592,
40749289289289 = 63835172.
9282 = 163 + 563 + 883.
652 + 662 + 672 + ... + 9282 = 163322.
9282 = 861184, 8 + 6 + 1 + 1 + 84 = 102,
9282 = 861184, 86 + 1 + 1 + 8 + 4 = 102,
9282 = 861184, 82 + 62 + 12 + 182 + 42 = 212.
by Yoshio Mimura, Kobe, Japan
929
The smallest squares containing k 929's :
5929 = 772,
36929929 = 60772,
19296929051929 = 43928272.
The squares which begin with 929 and end in 929 are
9297394929 = 964232, 92919499929 = 3048272, 92978035929 = 3049232,
929147549929 = 9639232, 929444461929 = 9640772,...
9292 = 863041, a square with different digits.
Komachi square sum : 9292 = 142 + 382 + 762 + 9252.
the square root of 929 is 30. 4 7 9 5 0 13 0 8 2 5 6 3 4 0 9 5 8 10 8 6 6 ...,
and 302 = 42 + 72 + 92 + 52 + 02 + 132 + 02 + 82 + 22 + 52 + 62 + 32 + 42 + 02 + 92 + 52 + 82 + 102 + 82 + 62 + 62.
3-by-3 magic squares consisting of different squares with constant 9292:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(7, 96, 924, 444, 812, 81, 816, 441, 52), | (20, 465, 804, 615, 596, 360, 696, 540, 295), |
(24, 153, 916, 601, 696, 132, 708, 596, 81), | (25, 396, 840, 504, 700, 345, 780, 465, 196), |
(48, 396, 839, 596, 657, 276, 711, 524, 288), | (57, 144, 916, 384, 839, 108, 844, 372, 111), |
(57, 304, 876, 564, 708, 209, 736, 519, 228), | (84, 273, 884, 312, 844, 231, 871, 276, 168), |
(84, 592, 711, 623, 564, 396, 684, 441, 448), | (96, 196, 903, 252, 879, 164, 889, 228, 144), |
(96, 385, 840, 560, 696, 255, 735, 480, 304), | (111, 452, 804, 636, 624, 263, 668, 519, 384), |
(164, 552, 729, 636, 601, 312, 657, 444, 484) |
9292 = 863041, 8 + 6 + 30 + 4 + 1 = 72,
9292 = 863041, 86 + 30 + 4 + 1 = 112.
by Yoshio Mimura, Kobe, Japan